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A111863 a(n) is the smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.

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%I A111863 #21 Jan 19 2023 11:09:43
%S A111863 5,11,17,23,29,5,41,47,53,59,5,71,11,83,89,5,101,107,113,17,5,131,137,
%T A111863 11,149,5,23,167,173,179,5,191,197,29,11,5,17,227,233,239,5,251,257,
%U A111863 263,269,5,281,41,293,23,5,311,317,17,47,5,11,347,353,359,5,53,29,383,389,5
%N A111863 a(n) is the smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.
%C A111863 From _Robert Israel_, Jan 18 2023: (Start)
%C A111863 a(n) = 5 if n == 1 (mod 5).
%C A111863 a(n) = 6*n - 1 if n is in A024898. (End)
%D A111863 G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 2, Section 2, Problem 96.
%H A111863 Robert Israel, <a href="/A111863/b111863.txt">Table of n, a(n) for n = 1..10000</a>
%e A111863 For n = 13, 6*n - 1 = 77 = 7*11; 7 == 1 (mod 6), but 11 == 5 (mod 6), so a(13) = 11.
%p A111863 f:= n -> min(select(p -> p mod 6 = 5, numtheory:-factorset(6*n-1))):
%p A111863 map(f, [$1..100]); # _Robert Israel_, Jan 18 2023
%o A111863 (PARI) for(k=1,60,my(f=factor(6*k-1)[,1]);for(j=1,#f,if(f[j]%6==5,print1(f[j],", ");break))) \\ _Hugo Pfoertner_, Dec 25 2019
%Y A111863 Cf. A007528, A016969, A024898, A107745.
%K A111863 nonn,look
%O A111863 1,1
%A A111863 _Reinhard Zumkeller_, Aug 20 2005